There are likely more than 10.000 known rules for card games. I would guess, that at least 95% of them have some practical use of the suit system, maybe exotic games like "memory with cards" or "Schwarzer Peter" don't use them, but only identical pairs. Counting games possibly use only the numbers independent from the suits.

I don't know a card game, which uses the 1-3-6-10-15-21 system. Do you know one?

Readers of this thread may be interested in Moakley's earlier ideas, presented in her 1956 paper "The Tarot Trumps and Petrarch's Trionfi" in the Bulletin of the New York Public Library, as well as her correspondence with Erwin Panofsky, from 1955 to 1967. I have both of these thanks to the efforts of the late Michael Hurst.

The data totals 31 megs. I'd be happy to send them to anyone, like Mike or Huck, who could post them here. I'm not sure how to do it.

There are likely more than 10.000 known rules for card games. I would guess, that at least 95% of them have some practical use of the suit system, maybe exotic games like "memory with cards" or "Schwarzer Peter" don't use them, but only identical pairs. Counting games possibly use only the numbers independent from the suits.

I don't know a card game, which uses the 1-3-6-10-15-21 system. Do you know one?

?

Moakley considers 21 as a Triangular number
She considers 56 as a Pyramidal number.

She links these numbers to dices games.

Nevertheless I noted that stricto sensu 56 would be a tetraedral pyramidal number that is : 1+3+6+10+15+21
But the 56 are not a serie such as the 22(see NB)
The 56 are to be divided in 4.
That is 14x4
In the Arithmological structure, the 56 correspond to
(40 : 4) + (16 : 4)

BOUGEAREL Alain wrote:
Moakley considers 21 as a Triangular number
She considers 56 as a Pyramidal number.

She links these numbers to dices games.

Nevertheless I noted that stricto sensu 56 would be a tetraedral pyramidal number that is : 1+3+6+10+15+21
But the 56 are not a serie such as the 22(see NB)
The 56 are to be divided in 4.
That is 14x4
In the Arithmological structure, the 56 correspond to
(40 : 4) + (16 : 4)

NB
The 22 are a serie corresponding to the Pentagonal Number 22 = 1+4+7+10

Moakley is correct, when she links 21 and 56 to (popular) dice games. There are 21 result possibilities for throws with 2 dice and 56 results for throws with 3 dice. There is also an influence on lot book systems. As dice games had been popular, it's a plausible suspicion, that this condition might have influenced the structure of Tarot. But that is naturally not secure, and I guess, there is no way to prove this ... or the contrary.

But what has this to do with the sequence 1+3+6+10+15+21 = 56?
It's a number theory curiosity, that here the number 21 and 56 also appear, but what game is influenced by this sequence, especially which card game? There are many number curiosities in number theory, and not all are under suspicion to have influenced anything on the playing card table.

Sure, you can interpret the 14 cards of a suit as 10+4 and you can associate this to the Tetraktys, if you like.
4x14 = 56.
But there are other decks with 4x13 and 4x12, which also were very popular (more than 4x14) and these don't work with a hidden tetraktys structure. The system "4 suits" and "somehow a number of cards in each suit" is a standard with enough freedom to create just decks in variants. And also the number of suits wasn't written in stone.

Huck
It seems I failed to make me understood by you at least. I don't know about others.

So let's come back to the beginning - that is Moakley statement about the these numbers 21 and 56 in Tarocchi.

Moakley had written :

"We have come a long way from the imaginary knights which make up the suit cards of the tarocchi, through the factual story of the family for whom the cards were made. But before we go on to the trump cards there is one more question to' consider. Why are there fifty-six suit cards, and why are there twenty-one trumps? The answer is found when we remember that cards, as a game of chance, replaced dice almost completely. In the dice games which use three dice, there are fifty-six possible throws, and with two dice twenty-one. There are other, more fanciful, considerations which make these numbers suitable. Twenty-one is a triangular number with a base of six that is, 6 + 5 + 4 + 3 + 2 + 1= 21. Fifty-six is a pyramidal number' with a base of twenty-one (a pyramid of fifty-six balls may be raised with the former triangle of twenty-one as its base). Add the "wild" Fool to get seventy-eight cards and you have another triangular number with a base of twelve. Take away the Fool and you have the product of seven and eleven, those numbers' symbolical of luck and dear to the dice player. (9)
_______________
[note originally p. 42]
9. It was Professor Maurice G. Kendall who pointed out to me that fifty-six is the number of throws with three dice. See Kendall ("Studies" p 1-14). He mentions the dice game of fifty-six throws which Bishop Wibold recommended to his clergy as a spiritual exercise in the year 970. Burckhardt (Civilization, p 409) mistakenly refers to this as a game of cards. The original source is Mon Germ SS. vii, p 433."

Point 1 : she is not speaking about 48 or 52 suits cards but 56 "suit cards of the tarocchi"
Consequence : your last argument misses it's objectf -"But there are other decks with 4x13 and 4x12, which also were very popular (more than 4x14) and these don't work with a hidden tetraktys structure. The system "4 suits" and "somehow a number of cards in each suit" is a standard with enough freedom to create just decks in variants. And also the number of suits wasn't written in stone."

Point 2 . she, not I, gives 21 - "Twenty-one is a triangular number with a base of six that is, 6 + 5 + 4 + 3 + 2 + 1= 21."

Point 3 : she, not I, gives 56 - "Fifty-six is a pyramidal number' with a base of twenty-one (a pyramid of fifty-six balls may be raised with the former triangle of twenty-one as its base)"
I, not she, explained that this is a tetrahedral pyramidal number that is : 1+3+6+10+15+21
To make it clearer, 21 at the basis, then, 15, then 10, then 6, then, 3, then 1.

Point 4 :
"Why are there fifty-six suit cards, and why are there twenty-one trumps? The answer is found when we remember that cards, as a game of chance, replaced dice almost completely. In the dice games which use three dice, there are fifty-six possible throws, and with two dice twenty-one."
She quotes in note Kendall who relates these two numbers to dices throws of 2 and 3 dices: "It was Professor Maurice G. Kendall who pointed out to me that fifty-six is the number of throws with three dice. See Kendall ("Studies" p 1-14). He mentions the dice game of fifty-six throws which Bishop Wibold recommended to his clergy as a spiritual exercise in the year 970. Burckhardt (Civilization, p 409) mistakenly refers to this as a game of cards. The original source is Mon Germ SS. vii, p 433"

Point 5 : I granted Moakley for taking in account in a pythagorean arithmetic in Tarocchi the Numbers 21 and 78 as Triangulars and 56 as "pyramidal" .
" There are other, more fanciful, considerations which make these numbers suitable. "
"Twenty-one is a triangular number with a base of six that is, 6 + 5 + 4 + 3 + 2 + 1= 21. Fifty-six is a pyramidal number' with a base of twenty-one (a pyramid of fifty-six balls may be raised with the former triangle of twenty-one as its base). Add the "wild" Fool to get seventy-eight cards and you have another triangular number with a base of twelve"

Point 6 : I nevertheless, wrote that :

A. She did not unearth the Pentagonal Number 22 as 1+4+7+10
B. She related Number 56 as a pyramidal number with a basis of 21 + 15 + 10 +6 + 3+ 1
In tarocchi we do not have such a progression but the tetrahaedral 56 is, in fact, divided in 4
That is : 4x14
The arithmological structure shows these two numbers as :
4 x (10 +4)

Point 7 : She saw, but others did before such as Etteila's disciple, that 78 was the Triangular of 12 :
"Add the "wild" Fool to get seventy-eight cards and you have another triangular number with a base of twelve."
Yet she did not see the complete structure of the 78 as on :

Point 8 :
You wrote, talking about 21 and 56, but I now finally refer not to Moakley's 77 (21+56) but to 78 as 22 + (40 +16) -with 40 as (4x10) and 16 as (4X4) : "what game is influenced by this sequence, especially which card game? There are many number curiosities in number theory, and not all are under suspicion to have influenced anything on the playing card table."
My hypothesis is precisely that the final state of the 78 cards of the Tarocchi were influenced by Pythagorean Number theory such the arithmological structure of the 78.http://letarot.it/page.aspx?id=603

BOUGEAREL Alain wrote:
Moakley ..
[note originally p. 42]
9. It was Professor Maurice G. Kendall who pointed out to me that fifty-six is the number of throws with three dice. See Kendall ("Studies" p 1-14). He mentions the dice game of fifty-six throws which Bishop Wibold recommended to his clergy as a spiritual exercise in the year 970. Burckhardt (Civilization, p 409) mistakenly refers to this as a game of cards. The original source is Mon Germ SS. vii, p 433."

Well, if Moakley did need Dr Kendall to realize, that there are 21 possibilities of two throws with dice, then she tells us, that she wasn't an expert in matters of games.
Also she wasn't an expert in lot books, cause these very often used the 56 in their systems ... cause a throw of 3 dice has so much possibilities.

Point 1 : she is not speaking about 48 or 52 suits cards but 56 "suit cards of the tarocchi"
Consequence : your last argument misses it's objectf -"But there are other decks with 4x13 and 4x12, which also were very popular (more than 4x14) and these don't work with a hidden tetraktys structure. The system "4 suits" and "somehow a number of cards in each suit" is a standard with enough freedom to create just decks in variants. And also the number of suits wasn't written in stone."

I spoke about 48 and 52 cards, and I had my own reasons, I needn't Moakley. This card deck structures (48, 52) were clearly more popular than the decks with 56 cards. I guess, that it is a reason. The most successful deck form in history is not Tarot, but the normal card deck.
Some people with a special focus on Tarot overlook this trivial fact.

Point 2 . she, not I, gives 21 - "Twenty-one is a triangular number with a base of six that is, 6 + 5 + 4 + 3 + 2 + 1= 21."

Point 3 : she, not I, gives 56 - "Fifty-six is a pyramidal number' with a base of twenty-one (a pyramid of fifty-six balls may be raised with the former triangle of twenty-one as its base)"
I, not she, explained that this is a tetrahedral pyramidal number that is : 1+3+6+10+15+21
To make it clearer, 21 at the basis, then, 15, then 10, then 6, then, 3, then 1.

There you understand, that the dice possibilities are the common object, not "pyramidal" or "triangular" numbers.

Alain, look at it with some distance ...

You play a little bit Tangram with the numbers 4x10 + 16 and then you had a rest and you called it the "mystery of the 22 special cards".
I guess, you wouldn't find it as a very representative picture of the number 22, if you would find it somewhere without the extension of 4x10+16.

You are fixed on the condition, that 78 = 1+2+3+4+5+6+7+8+9+10+11+12, but many Tarot/Trionfi versions didn't have 78 cards. Other types of decks haven't the feature, that they use a triangular number ... well, the Stubai valley Troggn uses 66 cards, a 52-cards+3 jokers has 55 cards and 36 cards are quite common normal decks. But is this really of interest ... ?

Well, we study playing cards and their historic development. Your internal figure of 22 elements isn't a historical figure. Or? There were many curious pictures and phenomenons in the past.

Deck designers were likely more interested to have an easy way to produce their cards, 78 = 6x13 isn't really a good choice under this aspect.

...
We'll, we were basically talking about Moakley...
Anyway, I agree we disagree about Pythagorean Number theory influence on Tarocchi's 78 final settle ...

...

Other considerations about the 56 :

A. "The 56-card deck ... was the basis for Tarot"

"Trick-taking games of some sort, traditionally the most popular form of card game, probably arrived with the cards. However, while we know almost nothing about the games played, the design of the Mamluk decks changed very little as adopted by the Italians, and so-called Moorish cards may have also been used in Europe. The four suits were Swords, Staves (the Mamluk decks used Polo Sticks), Coins, and Cups. Each suit had ten pip cards and three "court cards", a King, Knight, and Page, creating a 52-card deck. While that basic Italian suit system continued to be used, variations developed almost immediately. Spanish decks changed the Staves into Clubs, and altered the designs, and German cardmakers developed a number of alternative suit-systems.
In the earliest known description of playing cards, Brother John described decks in which the number of court cards and even the number of suits were increased from the norm, and some in which female figures were used on the court cards. In addition to regular decks, novelty decks were also produced in the 14th century, including one with images of gods and emblematic animals.
In "ordinary" cards, there are usually 13 : from 1 to 10 and 3 court cards
In "Tarots", there are usually 14 - from 1 to 10 and 4 court cards.

The suit-cards used in Tarot were the same as standard Italian playing cards. In some regular Italian-suited decks of the period, Queens had been added to the suit cards, creating a 56-card deck, and such a deck was the basis for Tarot'
(Cf : Taropedia http://www.tarotpedia.com/wiki/Tarot_History)

B. The most ancient card game of 56 with 16 Courts et 40 numérals would be a hand painted game dedicated to hunting .
"Ambras" (Germany)
Deck "Hofjagd", 1440/1445
handmade deck found in castle Ambras (Tirol) together with Hofämterspiel
reprinted by "Edition Leipzig" in 1971 for Heimeran (Munich)"http://trionfi.com/0/j/d/ambras/

C . Notes about the 56 throws of 3 dices with 6 faces :
"The 56 distinguishable occurences were associated with practise of Virtues for monacal reasons by Wibold of Cambrai circa 1000 AD.
He enumerated 56 virtues - one corresponding to each of the ways 3 dices could be thrown irrespective of order
The use of dices for the purpose of choosing among a number of possibilities is much older than Wibold.
The earliest approch to the counting of the numebrs of ways in which 3 dices can fall appears to occur in a latin poem "De vetula"; this remarquable work was regarded as Ovide's for some time, and is included in some of the medieval editions of his poems. It is however suppositious and several candadates have been proposed for authorship. The relevant passage may be briefly and freely construed as follows :
If all three numbers are alike, there are 6 possibilities.
If two are alike and the other different, there are 30 cases because the pair can be choosen in 6 ways
and if all three are different, there are 20 ways because 30 x 4 = 120, but each possibilities arises in 6 ways" [120 : 6 = 20]
Cf : https://books.google.fr/books?id=LLEZQC ... es&f=false

D. The number 56 divided in 4 in classical antiquity related to "divination" :

1.
"Two kinds of dice were used in classical antiquity: dice proper (kuboi, tessarae), which are virtually identical to six-sided modern dice, and knucklebones (astragali, tali), which have four sides (Halliday 205-15, esp. 213-15; David 1-7; Ore 193). For divination, five astragali were rolled, and the resulting combination was looked up on a four-sided pillar, many examples of which survive in more or less fragmentary form (e.g., Sterrett "Epig.", "Wolfe"; Kaibel). It so happens that the number of possible five-astragali throws is 56, exactly the number of Minor Arcana, and that the throws were listed on the tablets in four "suits" (two of 15 throws, two of 13; see Halliday 213n3)."
(http://wisdomofhypatia.com/OM/BA/PT/Mintro.html)

Halliday's reference given by Steve

quote p213

"From Asia
Minor we have inscriptions which contain a list of the possible throws and their interpretation. The system in the different inscriptions is identical, all are written in bad metre and indifferent Greek, the names of the throws are
constant in the various fragments, and the variants in the text itself are surprisingly few.
There is a fragment of a system of astragalomancy with seven astragali from Termessos,^ but the normal code is constructed for five astragali. Of this code we have fragments from Kosagatch, Tefeny, Yarishli, Sagalassos,
Termessos, Ordekji, Indjik, and Adalia.^ The astragalos has only four numbered sides with the values i, 3, 4, and 6. The totals, therefore, range from 1 + 1 + 1 + 1 + 1 = 5 ^o 6 + 6 + 6 + 6 + 6 = 30. Some of the intervening numbers,
e.g. 6 or 29, cannot be made out of combinations of I, 3, 4, and 6, but on the other hand many of the possible totals can be made by various combinations..., In all 56 throws are possible, and of these all except the 12th,
though some of them only in fragments, can be obtained from the various stones.^ The inscriptions seem to have been cut in columns, on the sides of a four-sided pillar. They give in the case of each throw the combination of figures and the total, followed by the name of the power to whom the throw belongs. The second line in some of the inscriptions consists of an attempt to force the numbers of the throw intometre. The meaning of the whole is given in three hexameters of very inferior quality. The second throw, for example, is : —

2.
You also noted :
"the old Greeks had another way to reach the number 56 in their lot book systems?
Five 4-sided dice were used, carrying the number 1-3-4-6.
forum.tarothistory.com/viewtopic.php?f=11&t=1171&p=19177#p19172"

"the 56 combinations in the old Greek lot oracle are from the times of Pythagoras."